Absorption machine irreversibility using new entropy calculations

Solar Energy Vol. 39, No. 3, pp. 243-256, 1987 Printed in the U.S.A.

0038-092X/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.

ABSORPTION M A C H I N E IRREVERSIBILITY U S I N G N E W ENTROPY C A L C U L A T I O N S D. K. ANAND'~and B. K U M A R : ] : University of Maryland, College Park, MD 20742, U.S.A. Abstract--This paper is concerned with the calculation of LiBr/H20 properties over the complete range of useful temperatures and concentrations applicable to the absorption cycle. This information is then used to calculate individual irreversibilities for all the components of single and double effect LiBr/H20 absorption cycles. In order to calculate the value of the specific entropy of aqueous LiBr, it is necessary to first calculate the enthalpy values. In addition, the activity coefficient of the salt for all concentrations at a given temperature must be known. The activity coefficient is not independent of the P-T-x data and can be obtained from it by carrying out the numerical integration of the GibbsDuhem equation. For consistency, in the calculation of entropy, it is necessary that the same P-T-x data be used both for the calculation of the enthalpy values, as well as to obtain the activity coefficient. The authors haveconsistently used the P-T-x data from Duhring equations to calculate the values of all thermodynamic properties. The values of the LiBr activity coefficients reported earlier and those calculated by the authors are very close to each other, but there is a systematic difference in that the calculated values are higher. Although the difference is small, it can still add up during the process of numerical integration which is essential to calculate the specific entropy over a range.of concentrations. Consequently, a systematically higher value of solution entropy is reported. However, the trend of entropy curves is quite similar to earlier results. Because the values of specific entropy reported here are based on one constant arm the most recent P-T-x data, they will be useful for any future second law analyses involving aqueous-LiBr cycles. For the case of steady fluid flow through a component of a thermodynamic cycle, the irreversibility is given by

1. h N T R O D U C T I O N

The second law of thermodynamics distinguishes between the nature of work and heat by stipulating that in any cyclic process, the heat from a single source at a uniform temperature can never be completely converted into work whereas the work supplied to a cyclic process can always be completely converted into heat. This leads to the concept of the maximum available work and the definition of availability and irreversibitity. It can be shown that the quantity of heat dQ, delivered to a system at temperature T, while the system delivers the useful work dW,, causes an increase in the availability db of the system given by

If a number of fluid streams enter and exit the component, the last term can be evaluated by (Ab)s = ~

The quantity on the right-hand side of eqn (1) is the maximum possible increase in availability which takes place during the corresponding reversible incremental process. The irreversibility or the "lost work" during this process is defined as

where, for a general process, d / ~ O. t Professor. ~: Graduate student.

(3)

rn,b, - ~_, m j b j , l

(5)

j

where m~ and my denote the mass flow ratio for the /th incoming and the j t h outgoing material stream, respectively. Similarly, bt is the availability of the ith incoming stream and bj is the availability of the jth outgoing stream. For a component over which isothermal conditions prevail, eqn (4) can be rewritten as I=

(l-"~)Qtn-Wx-(Ab)s.

(6)

The expressions given above can be used to obtain the measures of irreversibilities (or, departures from ideal reversible conditions) in individual processes of thermodynamic cycles. A conceptual discussion of the steps involved in the first and second law analyses of therm.odynamic cycles has been discussed extensively in [1, 2]. Apart from a systematic methodology, the calculation of irreversibility requires a fairly detailed and consistent data base of thermodynamic properties. In the case of LiBr/ H20 this is not available.

243

244

D. K. ANASD and B. KUMAR

This paper is concerned with the calculation of LiBr/H20 properties over the complete range of useful temperatures and concentrations applicable to the absorption cycle. This information is then used to calculate individual irreversibilities for all the components of single and double effect LiBr/ H20 absorption cycles.

2. "I'HERMODYNAMICPROPERTIES OF AQUEOUS SOLUTIONS

A complete analysis of the above type requires extensive calculations using detailed data on the thermodynamic properties of each substance involved in the cycle. The degree of rigor desired in the analysis dictates the range of the required thermodynamic properties and the accuracy needed. At the minimum, the equilibrium properties, either in tabulations or in the form of correlations, must be available. For a more careful and detailed analysis, the effect of hardware configuration must be included and the knowledge of some transport properties, i.e. viscosity and thermal conductivity, might be necessary. Fortunately, the transport properties are not needed for simple first and second law analyses of an absorption cycle. The most desirable way to include the thermodynamic properties of a substance would be the formulation of a fundamental equation in any one of the four standard forms. An equivalent representation would be an equation of state together with ideal-gas state properties. Both the formulations have the advantage of built-in thermodynamic consistency. For water substance, such fundamental equations exist and are internationally agreed upon. Unfortunately, for the aqueous solutions of LiBr, there are no such widely accepted formulations. The thermodynamic properties of ammoniawater systems have been investigated in detail and the results are reported[l, 2, 4-101. Among these, the data reported by IGT[4] contain the most recent information. The only source of entropy tables is a paper by Scatchard et al.[5]. The properties of aqueous-LiBr solutions, carlier available in scattered form, were compiled together in an IGT report[ll]. Some of this data was published by ASHRAE[3] and later updated by McNeely[12]. The latter is considered to be the most current data. Some entropy tables for LiBrwater solutions have been calculated by Loewer[13]. These will be discussed in greater detail in the next section. Some additional thermodynamic property data are available in an NBS circular[14], in Lewis and Randall[15] and in International Critical Tables[16]. For aqueous LiBr, or, for any binary solution with only one volatile constituent (here, water), the measurement of the vapor pressure is the key to the evaluation of thermodynamic properties.

P-T-x The early P-T-x data for aqueous LiBr were presented in International Critical Tables[16] in 1928. Later in 1955, Pennington[17] obtained some results for higher concentration. Additional work was done by the Dow Chemical Company[18], Greeley[19] in 1959 and by McNeely et al.[20] in 1964. The last author also made a comparative evaluation of the previous vapor-pressure data. The ASHRAE paper[12] and the resulting P-T-x charts (Duhring charts) are the most current and agreed upon for use in industry. In the last reference, good agreement was found between the vapor-pressure data from different sources, except some deviation at higher concentration (LiBr > 50%). Pennington's[17] data was obtained by the "direct static method" which involves the direct measurement of vapor pressures. Greely[19] and McNeely[20] used the "dew-point" method in conjunction with steam tables. Enthalpy data The data on the heat of dilution for aqueous LiBr were published by Lange and Schwartz[21]. Some data also appeared in a NBS circular[14]. The latter source was used by the authors of the IGT bulletin[ll] to establish h-T-x charts for aqueous-LiBr solutions. In order to develop various isotherms, they made use of the isobaric heat capacity presented in the International Critical Tables[16] and a private communication obtained from Pennington at Carrier Corp. In 1939, Haltenberger[22] outlined a method to estimate the heat of dilution from P-T-x data. When McNeely[12] presented his Duhring charts, he used a similar procedure to verify the values reported by Lange and Schwartz[20]. He made use of the isobaric heat capacity data reported by Loewer[13] to establish isotherms in the h-x plane. These are the most recent enthalpy values currently in use for aqueous LiBr. Entropy data Currently, the only available source of aqueousLiBr entropy data is the doctoral thesis of Loewer[13]. For the purposes of our analysis, these were recalculated. The reasons for these recalculations and the procedure that was used is detailed in the next section.

3. ENTROPY CALCULATIONS FOR AQUEOUS LIBr SOLUTIONS

From the point of view of thermodynamic compatibility, the best way to estimate the specific entropy of a substance is to use a fundamental equation. Unfortunately, fundamental equations are not available for the aqueous solutions of LiBr and an alternative must be found. A particularly desirable

245

Absorption machine irreversibility method would be one making use of the available and compatible enthalpy data for each solution. The range of concentration of the LiBr solutions which is encountered in practical absorption cycles can be as high as 0.6-0.7 fraction basis. This is by no means a dilute solution. Consequently, the equations for diluted solutions may not be applicable. The procedure to calculate the specific entropy of aqueous-LiBr salts by making use of available property data is outlined in this section. From first principles, and using the relations for the Gibbs' free energy and Helmholtz function for binary systems, the entropy of an aqueous solution can be derived as

P-T-x data

vapor to the fugacity in pure water state /*

al = f l f l .

(13)

However, in the range of pressures encountered the fugacity can be replaced by the absolute pressure

a~ = (P~IP'~).

(14)

The value of~/_~ has been investigated earlier. It is related to the activity a2 of the solute as follows: a2 = my 2 = n(m±"/±) 2.

(15)

For a binary solution, the two activities a l and a2 are not independent but related by the GibbsDehem equation + R {L7 r (lna, \ M,

In a2'~ d x ) ~/'~'/ • (7)

The data for the specific enthalpy of the solution and the relationships for its specific volume are available from McNeely[12] and IGT[ll] and can be used to numerically evaluate the first two integrals. The last term on the right-hand side, however, is recognized as the entropy of mixing (A Smi~). The procedure for evaluating it is outlined in Rant[23], Bosnjakovic[24] and Haase[25]. It is also explained by Loewer[13]. The principal steps are as follows: Asmix = (I -

x)Asl + xms2,

Asi= (q-~-R, lna,),

(8) (9)

s2=s2o-2Rzln(m-_'y±)+ ( ~ ) , ( I 0 ) S2o = lim s2,

(11)

x~0

and, for ideal solutions

AS =

--R(l|

I --I-

112)

[x In x + (1 -- x)ln(l -- x)].

(12)

From the above, it is clear that in order to estimate the specific entropy of the aqueous LiBr, the specific entropies of the water (available from steam tables), of solid LiBr (available from Latimer[26] and the enthalpy data should all be known. In addition, the "activity" a~ of the water in the aqueous solution and the "mean ionic activity coefficient" ~,~ of the LiBr ions in the electrolytic solution must be known. The "activity" a~ of the solvent (water) can be expressed as the ratio of the fugacity of the water

n, d In al +

n2

d In

a2

=

0.

(16)

This is strictly true when both the pressure and the temperature are constant. However, for solutions, the pressure alone has little effect on the activity coefficient and constant temperature is a sufficient criterion for equation (16) to apply. Therefore, it is standard procedure (e.g. Robinson and Stokes[27]) to experimentally measure the activity of one constituent (the solvent or solute) and estimate that of the other constituent by the GibbsDuhem equation. Thus, it is possible to estimate ",/± either by direct experiments (usually by measuring the potential of suitable cells with or without liquid junction) or, indirectly, by knowing the activity of the solvent (which can be found either from the vapor pressures or from changes in the freezing or boiling points). The vapor pressure, as stated earlier, can be either measured directly (the "direct static" method) or by successively passing it through desiccants (the "dynamic" method) or by letting the solution come into equilibrium with a reference solution (the "isopiestic" method), among others. At low concentrations, it is possible to theoretically predict the activity coefficients of the solute by the limiting law of Debye-Hfickel (discussed in many books, including Lewis[15]). This is convenient because the numerical integration of the Gibbs-Duhem equation is difficult for very low concentrations. At times, the ~/~ data from e.m.f, measurements is used to overcome this difficulty instead of the theoretical value. In 1947, Robinson and McCoach[28], published data on the values of ~/= in the form of a technical note. It was also published as part of a more comprehensive paper[29] in 1949 and later incorporated in their book published in 1958. These were the values used by Loewer[13] in calculations of entropy values. The above data on ~___of aqueous LiBr were ob-

246

D. K. ANAND and B. KUMAR

tained by using the Gibbs-Duhem equation on the vapor-pressure data obtained from isopiestic measurements. The vapor pressures themselves were not reported directly. In order to maintain thermodynamic consistency, the activity coefficients -/_~ of the solute should be measured from the same vapor-pressure data that is used to estimate the activity of the solvent. With this consideration, the most recent vapor-pressure values reported by McNeely[12] were used in conjunction with the Gibbs-Duhem equation to recalculate the values of ~/_~. Because the vapor pressures at lower concentration are not significantly different as reported by different sources, the value of ~,_* at the molality of 2 was matched with that .reported b y R o b i n s o n and McCoach[28]. (The authors had verified the value of ~/_- at this concentration from other sources.) While the actual results will be presented in the next section it is sufficient to point out here that the recalculated values of 3'-* differ significantly from those reported by Robinson et al.[27-29] and used by Loewer[13], in his calculations. Therefore, it was considered necessary to recalculate the entropy tables of aqueous LiBr by using the most recent P-T-x data, the most recent enthalpy data of McNeely[12] and the revised ~/± data, in eqns (7)(12). The results of these calculations are presented in the next section. 4. RESULTS

The results of the analysis are presented as follows:

• Results of property value calculations • Results of second law analyses. Results o f property vahw calculations As stated earlier, the only property value that needed to be recalculated for this work was the specific entropy of the aqueous-LiBr solution. However, it is necessary to incorporate the most recent P-T-x data and the corresponding specific enthalpies in order to obtain thermodynamically consistent values of specific entropy. The Duhring coefficients A and B supplied by McNeely[12] were used to generate P-T-x equations for this work. McNeely[12] had used a method proposed by Haltenberger[22] to generate h-T-x data for aqueous-LiBr solutions. This method is explained in detail in [12] and involves the use of the following: • P-T-x data • Clausius Clapeyron equation to obtain the latent heat of solution of aqueous LiBr • Use of steam tables for water vapor data • Numerical integration of the partial enthalpy of water in solution, over the concentration range. In addition, the specific entha!py of the solution at the base concentration (50% LiBr) is also required. McNeely provided a simplified table of steam table equations and constants which are used here for the sake of consistency. The authors repeated the procedure suggested by Haltenberger[22] and explained by McNeely[12] and recalculated the h-T-x curves. The results are given in Table 1 and also plotted in Fig. 1. The corn-

Table l.h-T-xtablesforaqueous-LiBrsolutions

Specific Enthalples of Aqueous LiBr Solutions (Btu/ib) X

r(OF)

.00

.10

.20

.30

.35

.40

40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 260.0 280.0 300.0 320.0 340.0 360.0 380.0 400.0

8.0 27.9 47.9 67.9 87.8 107.8 127.8 147.8 167.7 187.7 207.7 227.7 247.7

5.3 22.2 39.4 56.6 73.9 91.3 108.7 126.1 143.3 161.2 178.8 196.5 214.5 232.7 251.2 270.3 290.1 310.8 332.9

2.3 16.8 31.6 46.6 61.8 77.0 92.3 107.8 123.0 138.4 154.0 169.7 185.6 201.9 218.6 236,0 254.2 273.7 294.8

.0 12.4 25.2 38.3 51.6 65.0 78.4 91.9 105.4 119.0 132.6 146.4 160.5 174.8 189.5 204.8 220.9 238.0 256.7

-.7 10.8 22.9 35.1 47.6 60.1 72.7 85.4 98.1 110.8 123.6 136.6 149.6 163.0 176.7 190.8 205.6 221.3 238.2

-.7 10.0 21.3 32.8 44.5 56.3 68.2 80.1 92.1 104.0 116.0 128.1 140.4 152.7 165.4 178.3 191.8 205.8 220.7

267.7 287.7 307.8 327.8 348.0 368.2

.45

.50

.55

.60

.65

.70

.3 10.3 20.9 31.7 42.8 54.0 65.2 76.4 87.7 98.9 110.2 121.5 132.9 144.4 156.0 167.8 179.8 192.2 204.9

3.5 12.8 22.8 33.0 43.5 54.0 64.6 75.2 85.9 96.5 107.1 117.7 128.3 138.9 149.5 160.1 170.7 181.3 191.9

10.2 18.9 28.3 37.9 47.7 57.7 67.6 77.6 87.6 97.5 107.5 117.3 127.2 136.9 146.5 156.0 165.2 174.1 182.6

21.8 29.7 38.4 47.4 56.5 65.8 75.1 84.4 93.7 102.9 112.1 121.3 130.3 139.2

69.2 77.8 86.4 95.1 103.7 112.2 120.7 129.1 137.4 145.5 153.3 160.5 167.2 172.9 177.4

123.4 131.4 139.2 146.9 154.3 161.3 167.7 173.4 177.9 180.9

147.9 156.2 154.1 171.3 177.7

247

Absorption machine irreversibility

Specific Entholpy of Aqueous-Li Br (Btu/Ib)

T('F)

_

(Btu/Ib)

_ ~

400 320 240 160 8O 40

-I00

-200

I

0-2 X

i

I

I

0-4

0"6

0"8

!

1-00

(Fractlonal Concenlratratlon of Lithium Bromide]

Fig. 1. Enthalopyplots for aqueous-LiBrsolutions. parison between this data and McNeely's values[12] will be discussed in the next section. The authors carried out the numerical integration of the Gibbs-Duhem equation on the P-T-x data based on Duhring constants to generate the activity coefficient of LiBr. Above the molality of 1.8, the P-T-x data of McNeely are used. Below that value, the osmotic coefficients measured by Robinson and McCoach[28] yielded the P-T-x data. The calculated values of the activity coefficients (at 25°C), along with the values reported by Robinson and McCoach[28] are presented in Table 2. This comparison is plotted in Fig. 2. The procedure for enthalpy calculations discussed in detail in Loewer[13] was then carried out to recalculate the specific entropy values of aqueous LiBr. For this purpose, the P-T-x and hT-~ data based on the Duhring charts and the recalculated activity coefficients were used. A comparison of the specific entropy (at 25°C) with the values reported by Loewer[13] is shown in Fig. 3.

The computer algorithm created from the above procedure was used to generate the tables of specific entropy over the temperature range of 0-130°C and the concentration range of 0-70% LiBr. The results are shown in Table 3 and also plotted in Figure 4. Results o f second law analysis The equations presented in the previous section were used to calculate the results detailed in Tables 4-8. Table 5 shows the values of the mass flow rates and the thermodynamic properties (temperature, enthalpy and entropy) at various stages of the single effect LiBr/water absorption cycle. A schematic diagram of the corresponding single effect cycle is shown in Fig. 5. The sink temperature of 45°F is selected to match with the lowest temperature in the cycle (evaporator temperature). Table 6 shows the values for the cases of a double effect LiBr/ water cycle. The schematic diagram of the corre-

248

D . K . ANAND and B. KU~AR

Table 2. Comparison ofaqueous-LiBractivity coefficients at25°C (77°F) (Comparison o f the v a l u e s r e p o r t e d by Robinson and McCoach [28] wlth values calculated from Clbbs-Duhem equation)

Activity Coefficient (T±) Molallty m

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1,2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 I0.0 II.0 12.0 13.0 14,0 15.0 16.0 17.0 18.0 19.0 20.0

Robinson and McCoach [28]

Gibbs-Duhem Equation

0.796 0.766 0.756 0.752 0.753 0.758 0.767 0.777 0.789 0.803 0.837 0.874 0.917 0.964 1.015 1.161 1.341 1.584 1.897 2.28 2.74 3.92 5.76 8.61 12.92 19.92 31.0 46.3 70.6 104.7 146.0 198.0 260.0 331.0 411.0 485.0

0.796 0.766 0.756 0.753 0.754 0.759 0.767 0.778 0.790 0.804 0.838 0.875 0.917 0.965 1.015 1.282 1.469 1.752 2.062 2.465 2.89 4.34 6.46 9.40 13.85 21.57 33.70 50.11 74.59 107.1 149.6 200.3 260.0 328.1 398.0 474.0

Absorption machine irreversibility

Activity Coefficient

249

for Aqueous LiBr

6

5

_

hem

/

In (Y-+)

-oo.oo

/

O'

?'

Roblnson and McCoaeh ['28]

_

2___

0

o

5

~o

t~

2'o

~' m (molallty)

Fig. 2. Plots of aqueous-LiBr activity coefficients at 25°C (77°F).

250

D.K. ANANDand B. KUMAR

Table 3. s-T-x tables for aqueous-LiBr solutions Specific Entropy o f Aqueo.us LiBr Solution (kCallkg*C) T ---> (DEG. C) .0

20,0 3 0 . 0

I0,0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

110.0

120.0

130,0

.0000

•8114 .8473 .881g .9154 .9478 .9792

1.0096 1.0391 1.0678 1.0957 1.122~ 1.1492 1.1749 1.2000

.0500

.7940 .8271 .8592 .8901 .9200 .9490

.9771 1.0044 1,0309 1.0567 1.081/ 1.1061 1.129g 1.1530

.I000

• 7719 .8028 .8325 .8613 .8891 .9160

. 9 4 2 2 .9675

.9921 1.0161 1.0393 1.0620 1.0841 1.1056

.1500

.7468 .7755 .8032 .8299 .8558 .8809

. 9 0 5 2 .9288

.9517

.9740

.2000

•7~8

, 8 6 8 2 .8901

.9114

. 9 3 2 2 .9523

.9719

.9911 1.0097

.7475 .7732 .7981 .8222 .8455

.9956 1.0167 1.03/3 1.0573

.2500

•6932 .7181 .7420 .7652 .7876 .8093

. 8 3 0 4 .8508 . 8 7 0 /

.8899

.9087

.92/0

.9448

.9621

.3000

.6642 .6874 .7097 ./313

. 7 9 2 0 .8110

.8475

.8650

.88~

.8986

.914/

.7522 .7724

.3500

.6338

. 7 5 3 0 .7708 .7880

.8048

.8211

.83/0

.8524

.8675

.4000

•6017 .6219 .6414 .6603 .6785 . 6 9 6 2

. 7 1 3 3 .7300

.7461

.7618

.llll

.79~)

.8065

.8206

,4500

.5683

. 6 7 3 3 .6889

.7041

.7189

.7332 .7472

.7608

.7/41

.5000

.5346 .5525 .5697 .5864 .6026 .6182

. 6 3 3 3 .8481 .6623

. 6 7 6 2 .6891

.7029

.7157

.7282

.5500

.5025 .5192 .5354 .5510 .5661 .5008

. 5 9 4 9 .6887

.6221

.6351

.6478

.6681

.6721

.6838

. 5 5 8 7 .5714

.5838

.5959

.60/6

.6198

.6301

.6409

. 5 2 3 5 . 5 3 5 2 .5466

.5577

.5684

.5789

.5891

.5991

.5091

.5192 .5291

.5387

.5480

.6000 i

--

.6500

.

.7000

.

.6553 .6762 .6963 . / 1 5 8 .7347

.8295

.5873 .6057 °6234 .6406 .65/2

.4886 .5036 .5180 .5320 .5456 .

. .

. .

. .

.

. 4 8 6 1 .4990 .5114 .

.

.

.

.

.

.

.

.

.

.4880

.4987

Table 4. Tabulation of LiBr/water absorption cycle parameters

mhw

:

2,495 kg/hr

(5,500 Ib/hr)

mcw

:

2,720 kg/hr

(6,000 Ib/hr)

mchw

:

1,633 kg/hr

(3,600 Ib/hr)

UAgen

:

1,633 kCal/hr.°C

(3,600 Btu/hr.°F)

UAcw

:

3,265 kCal/hr.°C

(7,200 Btu/hr.°F)

UAchw

:

1,633 kCal/hr.°C

(3,600 Btu/hr.°F)

TGEN

:

87.8°C (190°F), for single effect cycle 140.6°C (285°F) and 87.8°C (190°F) for double effect cycle

T

COND

37.8°C (tOO°F)

TABS

37.8°C (tOO°F)

TEVAP

7.2°C (45°F)

Effectiveness of a11 solution heat exchangers:

0.7

251

Absorption machine irreversibility

Specific Entropy/ of Aqueous LiBr at 25°C

1.0

Recalculated L o e w e r Dotal 13] 0.8

S 0.6

"e 0.4

0.2

I

I

0.2

0

I

0.4

I

0.6

L

0.8

1.0

X(Wt. f r a c t i o n of LiBr in aqueous solutions) Fig. 3. Plots of aqueous-LiBr specific entropies at 25°C (77°F). Table 5. Tabulationsfora single effect LiBr/watercycle Summary of LiBr/Water Cycle (by section) Section

m

T

x

h

s

b

1 2 3 4 5 6 7 8 10 II 20

9.008 9.008 9.008 8.008 8.008 8.008 1.000 1.000 1.000 l.O00 154.740 154.740 168.808 168.808 168.808 101.285 101.285

I00.00 I00.00 151.00 190.00 127.20 127.20 190.00 I00.00 45.00 45.00 207.78 199.24 80.21 87.66 94.04 60.82 50.82

.55362 .55362 .55362 .62275 .62275 .62275 0 0 0 0 0 0 0 0 0 0 0

-71.51 -71.51 -46.48 -30.34 -58.49 -58.49 1145.74 67.68 67.68 1080.90 175.31 166.78 47.93 55.36 61.73 28.57 18.59

.560274 .560274 .600576 .575976 .533836 .533836 2.861557 .940697 .946228 2.953788 1.116482 1.103628 .904767 .918435 .930010 .868258 .848895

68.5634 68.5634 71.6409 101.1806 95.9842 95.9842 32.3046 0.537 -2.4759 -82.7736 12.4169 10.8885 0.3581 0.3441 0.4082 0.8846 1.4516

21 22 23 24 25 26

m T x h

: : : :

s b

: :

mass flow rate/refrigerant flow rate temperature (°F) weight fraction of LiBr specific enthalpy (Btu/ib) after converting to IGT [ll] datum specific entropy (Btu/Ib°F) specific availability (Btu/ib)

252

D K. ANANDand B. KU,~tAR Table 6 Tabulations for a double effect LiBrlwater cycle Summary o f LIBr/Water Cycle (by section)

Section

m

T

x

h

s

b

1 2 3 4 5 6

9.008 9.008 9.008 8.008 8.008 8,008 0.470 1.000 1.000 1.000 9.008 8.479 8.479 8.479 .530 .530 .530 154.740 154.740 168.808 168.808 168.808 101.285 101.285

100.00 100.00 150.86 190.00 127.20 127.20 190.00 I00.00 45.00 45.00 234.44 285.00 190.71 190.71 285.00 190.00 100,00 295.74 290.58 84.65 92.09 95.38 60.82 50.82

.55362 .55362 .55362 .62275 .62275 .62275 0 0 0 0 .55362 .58821 .58821 .58821 0 0 0 0 0 0 0 0 0 0

-71.51 -71.51 -46.48 -30.34 -58.49 -58.49 1145.74 67.68 67.68 1080.90 - 5.04 14.68 -29.35 -29.36 1186.76 158.14 158.14 263.22 258.06 52.35 59.78 63.06 28.57 18.59

.560274 .560274 .600575 .575976 .533836 .533836 2.861557 .940697 .946228 2.953783 .660022 .662295 .602809 .602809 2.661273 1.089532 1.102317 1.240048 1.233208 .912932 .926490 .932415 ,862258 .848895

68.5634 68.5634 71.6409 101.1806 95.9842 95.9842 32.3046 0.5370 -2.4759 -82.7736 80.7007 99.1826 87.5446 87.5446 182.4193 9.9266 2.9626 33.0205 31.5863 0.3306 0.3756 0.4282 4.1528 1.4516

7

8 10 11 13 14 15 16 17 18 19 20 21 22 23 24 25 26

m T x h

: : : :

s b

: :

mass flow rate/refrlgerant flow rate temperature (*F) LiBr weight fractlon specific enthalpy (Btu/Ib) after converting to IGT [II] datum specific entropy (Btu/Ib*F) specific availability (Btu/ib)

Table 7 Second law analysis results for a single effect LiBr/water cycle

Component

q kCal/kg (Btu/Ib)

Ab kCal/kg (Btu/Ib)

I kCal/kg (Btu/Ib)

Generator

734(1321)

109.6 ( 1 9 7 . 2 )

22.7 ( 4 0 . 8 )

Condenser

-599(-1078)

-17.7 (-31.8)

10

(18.1)

Evaporator

563(1013)

-44.6 (-80.3)

14

(25.3)

Absorber

-698(-1257)

-37.9 (-68.3)

38

(68.4)

S o l u t i o n Heat Exchanger

±125(±225)

-7.8 (-13.9)

7.8 ( 1 3 . 9 )

0

0

Pump

0

Throttles

0

Total

0

-

1.7 ( - 3 )

1.7 (3)

0

94.2 (169)

253

Absorption machine irreversibility

S p e c i f i c Entropy of A q u e o u s LiBr S o l u t i o n s

Fig. 4. Entropy plots for aqueous-LiBr solutions.

Table 8. Second law analysis results for a double effect LiBrIwater cycle

Component

q kcallkg (~tullb)

Ab kcallkg (~tullb)

Generator I Generator I1 Condenser Evaporator Absorber S o l u t i o n Heat Exchanger A S o l u t i o n Heat Exchanger B Pump Throttles Total

0

0

81.1 (146)

D. K. ANANDand B. KUMAR

254

I ( )

We,~$olut,on

Condenser

171

] Generator

(4) 4- ~.

(3)

..

Strong Solution ~ = ~ I ~ t'~l [.~ ~ l

Solution Heat changer

(6)=~

~.o

[ ~"t~_1

r~-45-F I : J r^-lOO'F~ "P°rat°r[ (,1) I ^~'°r~er I(')

-(2)

V"

Fig. 5. Single effect LiBr/water absorption cycle.

(21~ T~l-215"Y

4-.3) Solution Heat Exchanger

(24)

/ Q,

(23)

°A °

Weak w Solution

TG~TtgO'F

Tc -lO0"P Condenser

~,saetnttor-ll

(3) (16)

mrll(7 )

(4) '=t=.=-=.s-,-m" Strong Solution Solution Heat Exchanger °B °

(5}

T~-45"F

V••(10)

T^-IOO'r (I

Absorber

/0, Fig. 6. Double effect LiBr/water absorption cycle.

Absorption machine irreversibility sponding double effect cycle appears in Fig. 6. Cycle parameters are identified in Table 4. Based on the property values of Tables 1 and 3, the second law analysis o f a sfngle effect LiBr/water cycle is carried out and the results are shown in Table 7. As a check, the sum of individual heat inputs, work outputs and the availability increases over individual components must add up to zero (over the cycle) and they are seen to do that. The individual irreversibility values must be non-negative, as is evidenced. Table 8 contains results analogous to Table 7, except that the cycle is a double effect LiBr/water cycle.

5. DISCUSSIONS AND CONCLUSIONS

The P-T-x curves from the Duhring constants specified by McNeely[12] were successfully reproduced in the calculation o f the h-T-x data. McN e e l y ' s enthalpy versus temperature curve at the base concentration (50% LiBr), over the temperature range of 0-180°C was also reproduced. The same procedure (including the simplified steam table equations), was used to obtain good comparisons for the enthalpy of the solution over the 0 140°C temperature range. Above this temperature, at concentrations away from the base concentration (50% LiBr), a significant difference was observed between the solution enthalpy calculated and those reported by McNeely. The reason for this discrepancy is not known but it is suspected that the different schemes for numerical integration (graphical method based on 5% steps on the concentration scale, as used by McNeely, versus digital computer program based on a 1% step on the concentration scale, as used by the authors) may be the cause. Fortunately, the range of temperature concentration encountered for the first and second law analyses for the thermodynamic cycles here is such that there is no significant difference in the solution enthalpy from the two sources. In order to calculate the value of the specific entropy of aqueous LiBr, it is useful to first calculate the enthalpy values, which were obtained from the Duhring plots of P-T-x curves, as discussed earlier. In addition, one also needs to know the activity coefficient of the salt for all concentration at a given temperature. The activity coefficient is not independent of the P-T-x data and can be obtained from it by carrying out the numerical integration of the Gibbs-Duhem equation. The entropy values reported by Loewer[13] use the activity coefficient reported by Robinson and McCoach[28] which, in turn, was calculated by the use of the Gibbs-Duhem equation on their own PT-x data. In order to be consistent in the calculation of entropy, it is necessary that the same P-T-x data be used both for the calculation of the enthalpy values, as well as to obtain the activity coefficient. The authors have consistently used the P-T-x data from

255

Duhring equations[12] to calculate the values o f all thermodynamic properties. The values of the LiBr activity coefficients reported by Robinson and McCoach[28] and those calculated by the authors are very close to each other, but there is a systematic difference (Robinson's values are mostly lower than those calculated by the authors). Although the difference is small, it can still add up during the process of numerical integration which has to be done to calculate the specific entropy over a range of concentrations. Consequently, a systematically higher value of solution entropy is reported (as compared to the values of Loewer[13]). However, the trend of entropy curves is quite similar to the results of Loewer. Because the values of specific entropy reported here are based on one constant and the most recent P-T-x data, they will be useful for any future second law analyses involving aqueous-LiBr cycles. Acknowledgements--The early part of this work was supported by DOE Contract No. DE-AC03-79CS30204. Computer funds for this work were supported by the Computer Science Center at the University of Maryland.

NOMENCLATURE al a2

b

bi

b: (Ab)s ¢v

h I iz m mhw n'~cw mchw m± mi Inj tl! n2

P Pi

er Q q5 R R1

Rz $ Sw SLiBr S20

solvent activity solute activity availability (kcal/kg) availability of the "i"th incoming stream (kcal/ kg) availability of the "j"th outgoing stream (kcal/ kg) change in availability for cycle (keal/kg) control volume solvent fugacity (Pa) solvent fugacity in pure state (Pa) general notation for enthalpy (kcal/kg) general notation for irreversibility (kcal/kg) partial molal enthalpy of solute (kcal/kmole) partial molal enthalpy of solute at infinite dilution (kcal/kmole) molality mass flow rate of hot water (kg/hr) mass flow rate of cooling water (kg/hr) mass flow rate of chilled water (kg/hr) mean ionic molality mass flow ratio for the ith material stream entering the cycle mass flow ratio for thejth material stream leaving the cycle number of solvent moles number of solute moles general notation for vapor pressure (Pa) vapor pressure of solution (Pa) vapor pressure for pure solvent (Pa) heat input (kcal/kg) differential heat of dilution (kcal/krnole) universal gas constant (kcal/K kmole) gas constant for the solvent substance (kcal/K kmole) gas constant for the solute substance (kcal/K kmole) general notation for entropy (kcal/K kg) specific entropy of water (kcal/K kg) specific entropy of LiBr (kcal/K kg) partial molal entropy of LiBr at infinite dilution (kcal/K kmole)

256

D. K. ANAND and B. KUMAR

ASmix entropy of mixing (kcal/K kg) T temperature (K, °C) To ambient temperature (K, °C) TABS absorber temperature (°C) TCOND condenser temprature (°C) TEVAP evaporator temperature (°C) TGEN generator temperature (°C) UAcw heat transfer rate per unit temperature for the cooling water (kcal/hr °C) UAchw heat transfer rate per unit temperature for the chilled water (kcal/hr °C) UAg=n heat transfer rate per unit temperature for the generator hot water (kcal/hr °C) W u useful work (kcal/kg) Wx useful work over cycle (kcal/kg) X concentration of LiBr (percent weight) mean ionic activity coefficient REFERENCES

1. S. W. Briggs, Second law analysis of absorption refrigeration. AGA and IGT Conference on Natural Gas Research and Technology, Chicago, 1971. 2. D. K. Anand, K. W. Lindler, S. Schweitzer and W. J. Kennish, Second law analysis of solar powered absorption cooling cycles and systems. J. Solar Energy Engng 106, 291-298 (1984). 3. AS HRAE Handbook ofFundamentals, Chap. I,(1985). 4. Institute of Gas Technology, Research Bulletin 34 (1964). 5. G. Scatchard et al., Thermodynamic properties--Saturated liquid and vapor of ammonia-water mixtures. Refrig. Eng. 53, 413-419 (1947). 6. HVAC Guide, Vol. 31. (ASHVE, New York, 1953). 7. B. H. Jennings, New investigations in absorption refrigeration. Refrig. Eng. 30, 87-93 0935). 8. Kyes, F. G., Renaissance of absorption refrigeration cycle. Ind. Eng. Chem. 21,477-480 (1929).

9. A. B. Stickney, Graphs help to solve ammonia absorption system problems. Ref. Eng. 54, 451-57 (1947). 10. B. E. Eakin and R. A./vlacriss, ASHRAE Trans. 70, 319-327 (1964). 11. Institute of Gas Technology, Research Bulletin 14, The absorption cooling process. ITG, IIT, Technology Center, August 1957. 12. L. A. McNeeley, Thermodynamic properties of aqueous solutions of lithium bromide. ASHRAE Trans. 85, 413-434 0979). 13. H. Loewer, Ph.D. thesis, University of Karlsruhe (1960). 14. NBS Circular No. 500. 15. G. N. Lewis and M. Randall, Thermodynamics. McGraw-Hill, New York (1961). 16. International Critical Tables. McGraw-Hill, New York (1929). 17. W. Pennington, Refrig. Eng. 63, 57-61 (1959). 18. ACG Data Book, A-10-5, Dow Chemical Co. 19. E. M. Greeley, Carrier Corporation Report (1959). 20. L. A. McNeely et al., Carrier Corporation Report (1964). 21. E. Lange and E. Schwartz, Z. Physik Chem. 133, 129130 (1928). 22. W. Haltenberger, Ind. Eng. Chem. 31,783-786 (1939). 23. Z. Rant, Forsch. Ing. Wes. 26, 1 (1960). 24. F. Bosnjakovic, Tec.hnische Thermodynamik. Steinkopf-Verlag, Dresden (1937). 25. R. Haase, Thermodynamlcs of Irreversible Processes. Addison-Wesley, Reading, Mass. (1969). 26. W. Latimer, J. Am. Chent. Soc. 43, 818-826 (1921). 27. R. A. Robinson and R. H. Stokes, Electrolyte Solns., 1958. 28. R. A. Robinson and H. J. McCoach, J. Am. Chem. Soc. 69, 22-44 (1947). 29. R. A. Robinson and R. H. Stokes, Trans. Far. Soc. 45, 612-624 (I949).